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toisa wod () Consider the given differential equation on the interval -∞ < t < c. Assume that the members of a solution set satisfy the initial conditions. Do the solutions form a funda- mental set? 7. y" + 2ty' + t²y = 0, y₁ (1) = 2, y₁ (1) = -1, y₂ (1) = -4, y₂(1) = 2 O JI

toisa wod () Consider the given differential equation on the interval -∞ < t < c. Assume that the members of a solution set satisfy the initial conditions. Do the solutions form a funda- mental set? 7. y" + 2ty' + t²y = 0, y₁ (1) = 2, y₁ (1) = -1, y₂ (1) = -4, y₂(1) = 2 O JI

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Second and Higher Order Linear Differential Equations 2. y - y'= 0; y(0) = 4, y'(0) = 1, y"(0) = 3 y₁ (t) = 1, y₂(t)=e', y3 (t) = e 1 nov 3. y(4) + 4y" = 0; y(0) = 0, y'(0) = -1, y₁ (t) = 1, y₂ (t)=t, y3 (t) = cos 2t, 4. y" + 2y"=0; y(0) = 0, y'(0) = 3, y₁ (t) = 1, y₂(t)=t, y(t) = e 21 t>0; y(2) = 1, y(t) = t¹ 5. ty" + 3y"=0, y₁ (t) = 1, y₂(t) =t, 6. ty" + ty" - y = 0, t < 0; 11. y" + y₁ (t) = 1, 200LT 1703 STE Exercises 7-10: to jarr to anointiloa toisa Consider the given differential equation on the interval -∞ < t <∞o. Assume that the members of a solution set satisfy the initial conditions. Do the solutions form a funda- mental set? t y₂ (t) = ln(-t), y3 (t) = ť² 95mod omlashao, y(-1) = 1, y'(-1)=-1, y"(-1) = -1 7. y" + 2ty' + t²y = 0, y₁ (1)=2, y₁ (1) = -1, y₂ (1) = -4, y₂(1) = 2 8. y" + ty = 0, y₁ (0) = 0, y₁ (0) = 2, y₂ (0) = -1, y₂(0) = 0 9. y" + (sint)y = 0, y₁ (0) = 1, y₁ (0) = -1, y(0) = 0, y₂ (0) = 0, y2 (0) = 2, y3 (0) = 2, y3 (0) = -2, y(0) = 1 10089 10. y" + e'y" + y = 0, y₁ (1) = 0, y (1) = 1, y(1) = 1, y₂ (1) = 1, y (1) = 0, y3 (1) = -1, y3 (1) = 0, y(1) = 0 12. NOH y" (0) = -4, y" (0) = 8 y4 (t) = sin 2t y" (0) = -8 y'(2) = -, y" (2) = 0.0 x+y=0 14. y(4) - y" + y = 0 Exercises 16-19: Find a fundamental set {₁,₂} satisfying 16. y" - y = 0, y₁ (0) = 1, y₁ (0) = 0, 17. y" + y = 0, 3₁(0) = 1, y₁ (0) = 1, Exercises 11-15: The given differential equation has a fundamental set of solutions whose Wronskian W(t) is such that W(0) = 1. What is W(4)? t y" + y+y=0 15. (t² + 1)y" - 2ty" + y = 0 y₁(0) = 0, 13. y"+y" + ty = 0 the given initial conditions. 3₂(0) = 0, ₂(0) = 1, y₂ (1) = -1, ₂(0) = 1 ₂(0) = -1
Transcribed Image Text:Second and Higher Order Linear Differential Equations 2. y - y'= 0; y(0) = 4, y'(0) = 1, y"(0) = 3 y₁ (t) = 1, y₂(t)=e', y3 (t) = e 1 nov 3. y(4) + 4y" = 0; y(0) = 0, y'(0) = -1, y₁ (t) = 1, y₂ (t)=t, y3 (t) = cos 2t, 4. y" + 2y"=0; y(0) = 0, y'(0) = 3, y₁ (t) = 1, y₂(t)=t, y(t) = e 21 t>0; y(2) = 1, y(t) = t¹ 5. ty" + 3y"=0, y₁ (t) = 1, y₂(t) =t, 6. ty" + ty" - y = 0, t < 0; 11. y" + y₁ (t) = 1, 200LT 1703 STE Exercises 7-10: to jarr to anointiloa toisa Consider the given differential equation on the interval -∞ < t <∞o. Assume that the members of a solution set satisfy the initial conditions. Do the solutions form a funda- mental set? t y₂ (t) = ln(-t), y3 (t) = ť² 95mod omlashao, y(-1) = 1, y'(-1)=-1, y"(-1) = -1 7. y" + 2ty' + t²y = 0, y₁ (1)=2, y₁ (1) = -1, y₂ (1) = -4, y₂(1) = 2 8. y" + ty = 0, y₁ (0) = 0, y₁ (0) = 2, y₂ (0) = -1, y₂(0) = 0 9. y" + (sint)y = 0, y₁ (0) = 1, y₁ (0) = -1, y(0) = 0, y₂ (0) = 0, y2 (0) = 2, y3 (0) = 2, y3 (0) = -2, y(0) = 1 10089 10. y" + e'y" + y = 0, y₁ (1) = 0, y (1) = 1, y(1) = 1, y₂ (1) = 1, y (1) = 0, y3 (1) = -1, y3 (1) = 0, y(1) = 0 12. NOH y" (0) = -4, y" (0) = 8 y4 (t) = sin 2t y" (0) = -8 y'(2) = -, y" (2) = 0.0 x+y=0 14. y(4) - y" + y = 0 Exercises 16-19: Find a fundamental set {₁,₂} satisfying 16. y" - y = 0, y₁ (0) = 1, y₁ (0) = 0, 17. y" + y = 0, 3₁(0) = 1, y₁ (0) = 1, Exercises 11-15: The given differential equation has a fundamental set of solutions whose Wronskian W(t) is such that W(0) = 1. What is W(4)? t y" + y+y=0 15. (t² + 1)y" - 2ty" + y = 0 y₁(0) = 0, 13. y"+y" + ty = 0 the given initial conditions. 3₂(0) = 0, ₂(0) = 1, y₂ (1) = -1, ₂(0) = 1 ₂(0) = -1
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